- #1

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y = f(x1, x2, x3,... , xn)

xi has an error of ei, where 1 <= i <= n.

How can I calculate the error of y in terms of e1,...,en?

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- #1

- 64

- 0

y = f(x1, x2, x3,... , xn)

xi has an error of ei, where 1 <= i <= n.

How can I calculate the error of y in terms of e1,...,en?

- #2

Science Advisor

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It depends explicitly on the functional form of f as well as any dependencies among the X_{i}.

- #3

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It depends explicitly on the functional form of f as well as any dependencies among the X_{i}.

So what is this dependancy, is there a general formula?

For instance, how do you calculate,

p = v . i (errors: e

R = v / i (errors: e

errors of "p" and "R"?

- #4

Science Advisor

Homework Helper

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If the errors in x_{1}, x_{2}, ..., x_{n} are e_{1}, e_{2}, ..., e_{n}, respectively, then the error in f(x_{1}, x_{2}, ..., x_{n}) is [itex]E(f)= \frac{\partial f}{\partial x_1}e_1+ \frac{\partial f}{\partial x_2}e_2+ \cdot\cdot\cdot\frac{\partial f}{\partial x_n}e_n[/itex], approximately. (Approximately because we are using the derivative rather than the actual difference- but that will give a good upper bound on the possible error.)

In particular, if O(x,y)= x+ y then [itex]e_o= 1(e_x)+ 1(e_y)= e_x+ e_y[/itex], if P(v, i)= vi, then [itex]e_p= i e_v+ v e_i[/itex] and if P(v, i)= v/i= vi^{-1}, then [itex]e_R= i^{-1} e_v- vi^{-2}e_i[/itex]

Notice that

[tex]\frac{e_P}{P}= \frac{i}{vi}e_v+ \frac{v}{vi} e_i= \frac{e_v}{v}+ \frac{e_i}{i}[/tex]

and

[tex]\frac{e_R}{R}= \frac{e_v}{v}- \frac{e_i}{i}[/tex]

illustrating an old mechanics "rule of thumb": if measurements are added, their errors add and if errors are multiplied, their**relative** errors add.

In particular, if O(x,y)= x+ y then [itex]e_o= 1(e_x)+ 1(e_y)= e_x+ e_y[/itex], if P(v, i)= vi, then [itex]e_p= i e_v+ v e_i[/itex] and if P(v, i)= v/i= vi

Notice that

[tex]\frac{e_P}{P}= \frac{i}{vi}e_v+ \frac{v}{vi} e_i= \frac{e_v}{v}+ \frac{e_i}{i}[/tex]

and

[tex]\frac{e_R}{R}= \frac{e_v}{v}- \frac{e_i}{i}[/tex]

illustrating an old mechanics "rule of thumb": if measurements are added, their errors add and if errors are multiplied, their

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- #5

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Those are some great formulas, thank you very much.

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